Electronic excited states in conjugated polymers

,~i'nthctic Metals. 28 (1989) D463-D468

[)463

ELECTR,0NIC EXCITED STATES IN CONJUGATED POLYMERS

S. MAZUMDAR* Center for Nonlinear Studies, Los Alamos National Laboratory, Los Alamos, NM 87545 (USA) S. N. DIXIT Lawrence Livermore National Laboratory, Livermore, CA 94550 (USA)

ABSTRACT We re-examine the subgap neutral and charged soliton absorptions in polyacetylene and the 2aAg state in finite polyenes within the context of a simple Peierls-Hubbard model. Our motivation is to determine the origin of the optical gap in trans-polyacetylene and to determine the magnitude of the effective on-site Coulomb repulsion. We conclude that this term lies between 2.25t0 and 2.75t0, where to is the nearest neighbor one-electron hopping integral.

INTRODUCTION The role of electron correlation in conjugated polymers continues to be a controversial topic [1]. Experimental signatures of electron correlations axe seen in, (i) the negative spin densities in solitons in polyacetylene [2-4], (ii) the occurrence of the 21A9 state below the optically allowed liB,, state in finite polyenes [5], and (iii) the shifts of the neutral and charged soliton absorption frequencies from what is predicted on the basis of single particle theory, - within the SSH model both neutral and charged solitons are expected to absorb at a frequency half the optical gap of the uudoped polymer, in contradiction to what is observed experimentally [6,7]. Our motivation here is to obtain semiquantitative information from these data, i.e., we want to determine the relative contributions of the electron correlation and bond alternation to the optical gap. Specifically, we are interested in determining whether conjugated polymers are closer to being a Peierls semiconductor with electronelectron interactions playing a perturbative role, or whether they are Mott-Hubbard semiconductors with the optical gap dominated by correlations. Even if the argument is made that these systems lie in the "difficult regime of intermediate correlations" such classification is useful. This is because most conjugated polymers have complicated enough backbones that only simple band calculations are possible. On the other hand, Mott-Hubbard semiconductors have spectra that are very different from what would be predicted on the basis of band theory. Simple classifications can therefore lead *Permanent address: Department of Physics, University of Arizona, Tucson, AZ 85721 0379-6779/89/$3.50

© Elsevier Sequoia/Printed in The Netherlands

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to qualitative reasonings and educated guesses that may be the only route to understanding the more complicated polymers. Furthermore, such qualitative reasonings can lead to new theoretical approaches. Elsewhere it has been discussed [1] why it is difficult to get a proper estimate of the electron correlation from the spin density ratios in polyacetylene. This is because the average ratio of negative to positive spin densities is a strong function of the width of the neutral soliton. Thus unless the experimental width is precisely known, and unless the theoretical calculation is for the same chain length, comparison of theoretical predictions and the experimental ratio is meaningless. To complicate things further, the theoretical width of the spin density wave profile and the width of the soliton can be very different for nonzero electron correlations [8]. Most Haxtree-Fock calculations of spin density ratios neglect both these features and there is no way to judge the merit of these calculations. An important point, however, is that recent experiments [4] find the ratio of average negative to positive spin density to be much larger than what was originally claimed by Thomann et ai [2]. It is interesting that the recent figure of 0.42 is close to what was found by Soos and Ramasesha [9] within the PariserParr-Pople ( P P P ) Hamiltonian (although, again, it must be said that the P P P figure of 0.45 was for a soliton width nearly twice that estimated by Mehring et al [4], who have determined the new spin density ratio). Because of all of the above complications, we focus on the charged and neutral soliton absorptions and the 2lAg state.

TItE MODEL Since we axe interested primarily in answering a specific question, - what the origin of the optical gap is, we have to minimize the number of parameters. We are therefore interested in the simple Peierls-Hubbard Hamiltonian

H = U Z n i l n i l + to Z [ 1 i

+ (-1)i/f]

[c+~,ci+l,a+ c++l,~ci,]

(1)

i,q

It should be made clear at the outset which kinds of properties can or cannot be explained within Eq. (1), with the same uniform U. Bond alternation is enhanced by the intersite Coulomb correlations [10] that have been neglected within Eq. (1), while optical gaps and spin densities axe reduced by such terms. Therefore, no single U can explain all of these observations quantitatively. We therefore focus entirely on the various optical gaps. For an exactly or a nearly half-filled band a single correlation parameter is sufficient, at least for our purpose. We want to determine the minimum U within Eq. (1) that can reproduce the experimental behavior. It should also be made clear that the U we are interested in is exactly as defined in the microscopic Hazniltonian (1). In recent years the same notation "U" has been used to denote a variety of quantities (sometimes by the same authors), for e.g., as a perturbation parameter, as the shift in the neutral or charged soliton absorption energies from the midgap, etc. These quantities axe different from the microscopic parameter we are interested in.

D465 THE NEUTRAL AND CHARGED SOLITON ABSORPTIONS REVISITED We begin with t h e e x p e r i m e n t a l observation t h a t the neutral and charged solitons absorb at 1.4 ev at 0.45 ev respectively [6,7], instead of at 0.9 ev t h a t is predicted within t h e SSH model, the optical gap for t h e u n d o p e d polymer being 1.8 ev. Both m e a n field [11-13] and exact finite s y s t e m calculations [14-15] have dealt with the above features in polyacetylene. T h e m e a n field calculations a s s u m e at the outset t h a t the optical gap is d o m i n a t e d by bond alternation, and therefore there is no way to check this ab initio. Besides, even with the above a s s u m p t i o n such calculations predict a U t h a t is at the borderline of the critical value of this p a r a m e t e r where such approximations break down. T h e P P P calculations had actually predicted [14] the neutral soliton absorption at the band edge, b u t these also suffer from shortcomings. First, with a very large n u m b e r of intersite correlation p a r a m e t e r s , it is not possible to answer the question we are interested in, viz., whether the gap is d o m i n a t e d by correlations or bond alternation. Second, and more i m p o r t a n t , is the fact t h a t at the chain lengths investigated in these calculations, the neutral odd chain and the even chain absorb at nearly the same energy even at the ttuckel limit. T h u s while t h e prediction t h a t the neutral soliton should absorb at the band edge was a correct one, the actual numerical result was i n d e p e n d e n t of the correlation parameters. We have therefore repeated the exact numerical calculations, b u t we have eliminated the finite size effects by a simulation t h a t seems quite artificial at first. Our interest is to reach chain lengths where t h e odd chain absorbs at a frequency half the optical threshold frequency for t h e even chain, and t h e n ask w h a t h a p p e n s to t h e former gap at finite U. Since, however, all exact finite U calculations are limited to chain lengths where such calculations are not possible with realistic alternation parameters, we solve this problem by choosing very large ~t in Eq. (1), such t h a t the infinite chain behavior is s i m u l a t e d even at very short chain lengths. In the present case we have chosen (~ = 0.5, such that the optical gap for the N = 9 chain becomes nearly half t h a t of the optical gaps in N = 8 and 10 at U = 0. For t h e even chains we a s s u m e perfect b o n d alternation, b u t for N = 9 we have chosen the single site soliton configuration with the fourth and fifth bonds both as single bonds (note that N = 11 is therefore not suitable for such a calculation, while N = 13 h a s too large a Hamiltonian matrix). We are not interested in N ~ oo extrapolation, but are interested in answering t h e following question: since at U = 0 we now have R ~ 0.5 (where R = 2 A o / 2 A e , 2 A o , and 2A~ are the optical gaps of the odd and even chains), what is the m i n i m u m value of U at which t h e ratios R ° and R + are equal to the experimental values of (1.4/1.8) and (0.45/1.8) simultaneously? Here R ° = 2 A ° / 2 A e and R + = 2AO/2A~, where 2A°o a n d 2 A + are the optical gaps of t h e neutral a n d charged solitons respectively, t h e two energies being different for U ~ 0. We have therefore calculated the optical gaps in N~ = 8, N = 9, corresponding to the charged soliton; N e = 9 , N = 9, corresponding to the neutral soliton; and N e = 8, N = 8 and Ne = 10 and N = 10, the two even chains. IIere Ne is the n u m b e r of electrons on a chain of N sites. In Fig. 1 we have plotted 2 A e , 2 A ° and 2 A + , and against U/t for fixed 6 = 0.5. Note t h a t with this 6 not only 2Ao ~ ½ x 2A at U = 0, b u t also 2Ae is very nearly the N ~

o¢ value 4to6.

As seen in

Fig. 1, R ° and R + reach their experimental values within the shaded region corresponding to the range 2.25t0 < U < 2.75t0. A finite range is needed because we are interested in fitting two different quantitities simultaneously. We believe t h a t this is the U t h a t would be required to reproduce the optical features of t h e infinite chain, with electron-phonon interactions modifying it only slightly. C o m p a r i s o n s to other e s t i m a t e s are deferred until later.

Dg66 4 =0.5

\\

\\

I

o

~2

Ne=8, N=8 Ne=9, N=9-

ONe=8, N=9

1

2

I

I

3

4

5

U/t o

Fig. 1. Optical gaps as a function of

U/to of the

even chain with perfect bond alternation, and of the

neutral and charged odd chains in the single site soliton configuration.

TIIE 2lAg STATE From a recent measurement of the third order optical susceptibility in trans-polyacetylene [16] it has been claimed that the lowest two-photon state occurs at nearly the same energy as the allowed optical state, and that this implies U ~ to • While we cannot determine whether the state at 1.8 ev is really the lowest two-photon state, or whether this is the 2lAg state as has been claimed in the above work, we can determine the minimum U at which the 2lAg state crosses the l l B u state and appears below it. We do this by calculating the energies of the lowest excited states for all even chains 4 < N < 10. Thus once again our motivation is different from the existing P P P calculations. Due to space restrictions we present only the N = 6 and the N = 8 results in Fig. 2. (the N = 4 results may also be seen in reference 5, the N = 10 data will be presented elsewhere). What is important for our purpose is the following: while the higher lAg states (31Ag,41Ag etc) cross the llB~ at highly chain length dependent U, the crossing of the 2lAg and the l l B u occurs at a surprisingly chain length independent U ~ 2t0. This should be obvious even from the limited data of Fig. 2. The implication is then that this is true for all N , and even if the interpretation of Kajzar et al (i.e. the 2lAg and the l l B ~ occur at the same energy in polyacetylene) is correct, the minimum value of U is 2t0.

D467 ]

i

N=6, 8:0.1

o <]

,

2Ag 31Ag N = 8 , 5=0.1

T 0

I

1

,

2

3

2lAg

4

5

T

U/t o

Fig. 2. Tile location of the lowest excited states with respect to the ground state in N = 6 and N = 8 as a function of

U/to.

The 2lAg crosses the llBu at a chain length independent U.

GROUND STATE CORRELATIONS We have already indicated that a correct comparison between experimental spin density ratios and theoretical results is difficult. However, it is possible to compare different .theoretical models and ground state correlation functions to reach a single value of U. To illustratrate, Soos and Ramasesha have calculated the optical gaps in a large number of finite polyenes within the PPP-Ohno model and have demonstrated that the gas phase optical gaps in these systems are generated with remarkable accuracy. Instead of attemping to extrapolate to N ~ oo the optical gaps, we chose to compare ground state correlation functions within Eq. (1) and the P P P models. We chose only those correlation functions that are essentially determined by the short range coulomb parameter.

The reason for

proceeding in the above manner is that while the P P P model reproduces finite N excited states, the short range ground state correlation functions have already converged at these N. From consideration of several different chain lengths and boundary conditions we reach an effective on-site correlation parameter that also lies within the range given above. As discussed below, we therefore agree with one of the two conclusions reached from recent P P P calculations, viz., the optical gap in polyacetylene is dominated by correlation and bond alternation affects it only slightly [17,18]. We however do not agree that the effective U is simply the difference between the on-site and the nearest neighbor terms in the P P P Hamiltonian [17, 18]. This difference is just too low, and for models with slowly decaying long range potential the effective on-site term is larger. This indirectly agrees with Tavan and Schulten's calculation [17] of the N ---* oo optical gap from the exact Lieb-Wu expression of the same: the gap

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that is obtained for an effective U equal to the difference between the on-site and the nearest neighbor term in the PPP Hamiltonian is much smaller than what is expected from the extrapolation of the finite chain PPP data.

DISCUSSION From two different sets of experimental data and comparison to a theoretical model that has successfully reproduced the optical gaps of a a large number of polyenes we arrive at nearly the same range of U. As will be shown elsewhere, at the values of U we deem appropriate the optical gap is largely dominated by Coulomb correlation. This agrees with the conclusions by authors who have worked on the PPP model. Our estimate of U within Eq. (1) is only slightly smaller than that made by Baeriswyl and Maki [20], who from a completely different set of comparisons concluded that 2.4t0 < U < 3.2t0. The value suggested by Ovchinnikov et al [21] also lies close to the range suggested by us. It is interesting to note that the value for the dimensionless electron phonon coupling constant within the PPP model (0.11) is much closer to the "unbiased" value (0.08) obtained for this para meter by Baeriswyl than to the the SSH value (0.2). Mean field models and results therefore severely overestimate the role of the electron-phonon coupling.

REFERENCES 1. D. B~riswyl, D.K. Campbell and S. Mazumdar, review article to be published. 2. H. Thomann et al, Phys. Rev. Lett. 50 (1983) 533 3. S. Kuroda and H. Shirakawa, Svnth. Met. 17 (1987) 425 and references therein 4. M. Mehring, A. Grupp, P. H6fer and H. K~iss, Synth. Met. 28 (1989) D399 (these Proceedings) and references therein 5. B.S. Hudson, B.E. Kohler and K. Schulten, in Excited States, vol. 6 , edited by E.C. Lira (Academic Press, 1982). 6. B.R. Weinberger et ai, Phys. Rev. Lett. `5~ (1984) 86. 7. (a) J. Orenstein and G. L. Baker, Phys. Rev. Lett. 49 (1982) 1042 (b) Z. Vardeny and J. Taut, Phys. Rev. Lett. 54 (1985) 1844 8. H. Fukutome and M. Sasai, Progr. of Theor. Phvs 69. (1983) 373 9. Z.G. Soos and S. Ramasesha, Phys. Rev. Lett. 51 (1983) 2374. 10. S.N. Dixit and S. Mazumdar, Phys. Rev. 29 (1984) 1824. 11. K. R. Suhbaswamy and M. Grabowski, Phys. Rev. B24 (1981) 2168 12. W.K. Wu and S. Kivelson, Phys. Rev. B33 (1986) 8546. 13. J.C. Hicks and J. Tinka Gammel, Phys. Rev. B~7 (1988) 6315 14. Z.G. Soos and L.R. Ducasse, J. Chem. Phys. 78 (1983) 4092 15. Z.G. Soos and S. Ramasesha, Phys. Rev.B29 (1984) 5410 and references therein 16. F. Kajzar et al, Svnth. Met. 17 (1987) 563 17. P. Tavan and K. Schulten, Phys. Rev. B~(~ (1987) 4337. 18. Z. G. Soos and G. W. Hayden in Electroresnonsive Molecular and Polymeric Systems vol .1, edited by Terje A. Skotheim, Mercel Dekker (1988) 19. E.tt. Lieb and F.Y. Wu, Phys. Rev. Lett. 20 (1968) 1445. 20. D. Baeriswyl and K. Maki, Phys. Rev. B28 (1985) 6633. 21. A.A. Ovchinnikov, I.I. Ukrainskii and G.V. Kventsel, Soy. Phys. lJsn. 1,5 (1973) 575.